Local Systems in Groups

LOCAL SYSTEMS IN GROUPS

Dedicated to the memory of Hanna Neumann

J. A. HULSE

(Received 27 June 1972) Communicated by M. F. Newman

1. Introduction Throughout this paper we will assume that all groups are contained in some fixed but arbitrary universe. Thus the class £) of all groups becomes a set. If X is a class of groups then we assume that 1 e X and if H = Ge£ then H eX. German capitals will be used to denote classes of groups. We will use an extended form of the notation for closure operators on the set of all classes of groups introduced by Hall in [1]. If H is any set, let -^(H) be the set of all subsets of H.

DEFINITION 1.1. If ^£ 0>{H) for some set H, a map A: & ->• & is called an operator on & if X c A(X)S A(Y) for all X, Ye&> with X c Y. If also a is an ordinal number the operators A11 and A* on t? are denned by A°(X) = X, A\X) = A(u {A\X)\ P < a}) if a > 0 and A*(X) = u {A*(X) | a is an ordinal} for all lef. A is called a closure operator if A = A*. A relation of partial order may be defined on the set of all such operators on 0> by the rule that A ^ B if and only of A(X) £ B(X) for all Xe0». Then clearly A1 ^ Aa+i ^ A* for all ordinals a and if, for some a, A" = A'+1 then A" = Afi = A* for all ft 2: a. From this we see that an operator on ^ is a closure operator if and only if A = A2. Further since 8P is a set it is clear that there exists an ordinal a such that A" = A*+ * = A*. Thus it follows that A* is a closure operator and is the least closure operator B such that A ^ B. We will use small capitals to denote operators on the set of all classes of groups (in a given universe) and Roman capitals to denote operators on ^(^(K)) for some set K. In this paper we are concerned with operators derived from various types of local systems. 426

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DEFINITION 1.2. A set £ of subsets of a set H is called a local system for H if 0 ¥= & and each finite subset of H is contained in some member of 2.. J> is called weakly upper-directed if for each Q e 1 and each h e H there exists ReJ such that Q u {h} £ R and 2. is called a fower if J2 is well-ordered by inclusion.

DEFINITION 1.3. The operators L, LW, LU, L, are defined by GetX

LUX, LfX, respectively) if and only if G has a local system J of ^-subgroups (which is weakly upper-directed, upper-directed, a tower, respectively) for all

classes of groups X. The operators L,LW, Lu, Lt on ^(^(K)) are defined similarly with ^(^(K)) replacing the set of all classes of groups. We remark that in our definition of a local system we are following Hartley [2]. In [4] Kuros requires every local system to be upper-directed. It follows immediately from the definition, that

(1) Lt ^ Lu ^ Lw ^ L and hence for all ordinals a

a (2) h* g L"U ^ i£ S L and L? ^ L* ^ L£ g L* .

It is well known that L = L* and LUX = L*X if X — sX, where GesX if and only if G is a subgroup of some 3t-group. In [3] Hickin showed that Lu < L*. The first aim of this paper is to show that the inequalities in (1) and (2) are all 1 1 strict, with the exception that Lt* = L* and also that A" < A"" " < A* in the cases where A = L,, LU and LW with some restriction on the cardinality, a |, of the ordinals a. These results are given in

THEOREM A. Let cc be an ordinal number. Then: (i) i4 < L ; +1 (ii) if \a\ ^ Ko then L"W< L"W < L*; (iii) L* < L* and L" < L£; Xo +i (iv) (Hickin) if \a\ ^ 2 then L*u < L*a < L*; +1 (v) Lu $ t?, L^ < L^ and L* < L? < i*; (vi) Lt = C

Thus, in particular, the operators Lt, LU and LW are not closure operators and Lt* = L* < L* < L* = L. In Section 2 of the paper we prove some purely set theoretic lemmas required for the proof of Theorem A while in Section 3 we develop the necessary translation into group theory. This investigation was sparked off by the question as to whether the class

£s, the class of all groups G in which the set of all serial subgroups of G forms a complete sublattice of the lattice of all subgroups of G, is L-closed. This question

arose from the work of Hartley in [2] where he showed that fis contains all

locally finite groups. We have been unable to completely decide this question but we will prove the following results in Section 4.

THEOREM B. The class fis is Lw*-closed and if X = sX s £s then LX S £S.

We remark that no such local theorem can hold for the classes fin, fld or £a, consisting of all groups in which the set of all subnormal, ascendant or decendant subgroups, respectively, forms a complete sublattice of the lattice of all subgroups, since there exist locally nilpotent groups in which the join of two subnormal subgroups is neither ascendant nor descendant — see for example [7, Appendix D> Exercise 23]. If 3Jtvand 9K^ are the classes of groups with the minimum condition on all subgroups and all serial subgroups, respectively, then we will deduce the following extension of Theorem A of [2] as a corollary of Theorem B.

V V THEOREM C. 2RS <= £s and L2R <= £s. It is perhaps worth noting in contrast that an example in [2] shows that there are groups with the maximum condition on all subgroups which do not

lie in fis. 2. The set theory We begin this section with

LEMMA 2.1. For each ordinal oc and each set K with \K\^ max{X0,|oc|} there exists a subset 2. of 0>(K) such that 0) 0el; (ii) L/(J2) = L/(J) for all /?;

(iii) L/(J) cz LU"(J) = LU*(J2) if p < a.

PROOF, (iii) is just the contents of Lemma 1 of [3], (ii) is easily seen to follow

from the construction in [3] and (i) follows since LU(J u{0}) = LU(J) u {0} and Lw(i

LEMMA 2.2. // K is any set with \K\ ^ Ko then there exists a subset M of @(K) such that 0ei and

5 r PROOF. Let J " be the set of all finite subsets of K. Then \& \ = \K\ since K 5 is infinite. We now partition K into sets KF, one for each F e J ", with | KF | = | K | and let

3. = {0} u{FuKF\Fe^}, so that 0ei£ &>{K). Suppose now that Q, Re£\{0} and 2 £ R. Then Q = £ U K£ and i? == F U Kr for some E, F e 3?, so K£ £ F u KF and so,

since | F | < Ko ^ | KE \, KE n Kr # 0. Thus E = F and so Q = R. Hence it follows that any weakly upper-directed local system which is contained in St

must have at most one member distinct from 0, so LW(J) = 2. and so Lw*(.2) = £. However, 1 is a local system for X and K £ J and so 2. c L(J), as required.

LEMMA 2.3. // X is any set with \K.\ = Ka /or some non-limit ordinal a ^ 1 f/iew f/iere exists a set J ^ ^(X) such that 0ei and

PROOF. We are indebted to the referee for the following proof which is much shorter than our original. By Theorem 1 on p. 451 of Sierpinski [7] there exists a set ^ £ &>(K) such that

(1) |^|> Ka and \R\ = K, for all Ret% and (2) if S, Te@ and S ^ T then |S n T| < K^. Now clearly & also satisfies

(3) if S, TUT2,-- are countably many distinct elements of 3& then S $ (J"= I Tn.

For each xeK let 3tx = {Re&|xeR} and let R = {x eK | \@x\< KJ . Then | u {^x xe£} | ^ Ka and so by (1) we may replace K by K\R and M by ^ \ u {^ x e R) in order to assume

(4) \Stx\ ^ Ka for all xeK. Let M = {(x,n)|x6K, n = 0,1,2,-}.

Since j Af j = \K\ = Ka we may well-order M = {tp\P

a = i I I ft' I v e ?

and take «- II 9 •* — U -^n •

a Thus 0eJo £ J c ^ and Un

for K, giving X e Lw(^). But by (2) K $ @ and so 2 a LW(J2) .

It now remains to show that J = LU*(M). Let QuQ2e2 and gt £ Q2. Then there exist m, n with 0 ^ m,« ^ w and Xj, j, e K for 0 ^ i < m and 0 ^ j

such that

Si = U«-< and Q2 = (J; <„*',-

Then by (3) we must have m S n and xt = yt for 0 ^ i < m. Hence it follows

that LU(J) = 2 and so LU*(J2) = 2, as required. We remark that it is easy to see that if | K | ^ Xo and J £ ^>(X) then Lt(^) = Lu(2) = LW(J) and so we certainly require | K | i> Kt for Lemma 2.3 to hold.

3. The group theory

We begin with two lemmas, the first of which is proved in [3] except for the

part concerning the operator LW which follows easily.

No LEMMA 3.1. Let K be any set with \K\ = 2 . Then there exist groups

FQfor each Q £ K, where F0 = 1, such that the map

cf>: J <->£! = {G|Gs FQfor some QeJ}

is a bijection from the set of all 2. c 0> = 0>(K) with 0e M onto the set of all classes of groups contained in ^3. Further

for all ordinals a and all 2. with 06 J c 0>, and

Since, if Q # 0, the groups fQ are not finitely generated the argument used by Hickin will not work in full for the operators LW and L . Also, since it is easy to see that L,(^3) => ty, the method of [3] will not decide precisely where the sequence L"(Q) terminates but only give a lower bound. For these reasons we establish the following lemma which avoids these difficulties at the expense of a further restriction on the cardinality of K.

LEMMA 3.2. Let K be a set with \K\ = Ko. Then there exist groups AQ for each Q <= K, with A0 — 1, such that the map

: Jt^Q = {G\G S AQfor some Qe2] is a bijection from the set of all 2, £ & = &>{K) with 0e2 onto the set of all classes of groups contained in s$. Further

: Vt (J) f 4>: K(2) f-» 4(Q) and for all ordinals a and all 2 with

PROOF. Since \K\ = Ko we may assume that K is the set of all prime numbers. For each k e K let Ck be a cyclic group of order k. If Q £ K let AQ be the restricted direct product of the groups Ck with k e Q, so that A 0 = 1. We first show that ^} = L^} . Since s$ is contained in the class of locally finite Abelian groups so is L*)3. But if GGL^J and has a Sylow p-subgroup of order greater than p then G has a subgroup H of order p2 and so there exists a sub-

group B of G containing H and isomorphic with AQ for some Q. This is a contra- diction as the Sylow p-subgroup of B has order at most p. So if G e L^S then G is a locally finite Abelian group with Sylow p-subgroups of order at most p for each prime p. Thus Ge

We next show that if C ^ AQ and C ^ AR then gcR and C = AQ. For if p e Q then the Sylow p-subgroup of C has order p. But the Sylow p-subgroup

of AR has order p or 1 depending on whether p e R or p $ R. Now since C ^ AR it follows that Q ^ R and C = /1Q ^ ,4K. Hence if 3. is a set of ^-subgroups of AR then .2 c {^Q | Q c /?}. it is now easy to deduce that for each R £ K the map

is a bijection from the set of all local systems J for R with 2. <=, SP onto the set

of all local systems of ^-subgroups for AR which preserves the properties of being weakly upper-directed, upper-directed and a tower. Thus the lemma holds for a = 1 and the rest follows by induction. The corollary below follows immediately from Lemmas 2.1 and 3.2.

COROLLARY 3.3. Let a fee an ordinal with |a| ^ Ko. Then there exists a class of groups Qa such that L/(QJ = hJ(Qx)for all 0 and

for all y < a. Before proving Theorem A we establish one further lemma.

LEMMA 3.4. // for each ordinal a, Dx is the class of all groups of cardi-

nality at most Ka then LU(O0) = -O and Lt%00) = Da.

PROOF. Since a subgroup of a group G which is generated by a finite or

countable subset is at most countable, the set of all£)0-subgroups of G is an upper-

directed local system for G and so LU(O0) = O. We prove the second part by induction on a, the result being true by hypo-

thesis if a = 0. Suppose now that a > 0 and L,"(£>0) = fDe for all /? < a. Let

Ge£)a then by the induction hypothesis we may assume that |G = Ka and so G = {gy\yx}. If 6

G = u {Ge | 6 < coa}, so G e Lt"(C0) and so Oa £ Lt*(O0). Conversely if G e Lt"(O0) then there exist an ordinal y and subgroups Ge e u {i/(O0) | /? < a} such that Ge < G^ for all 9 < < y and G = U {G910 < y}. Thus by induction 101 <£ | G91 < X, for all 0 < y and so | y | ^ Ka.

Hence \G\ ^ |y|max{|Gel |0

PROOF OF THEOREM A. (i) L£ < L follows immediately from Lemmas 2.2 and 3.2. +1 (ii) C < Lw < L* for all a with |a| ^ Ko follows from Corollary 3.3. (iii) LU* < LW* follows from Lemmas 2.3 and 3.1 whence it is easy to deduce a that Lu < LW" for all ordinals a. a t+1 No (iv) Lu

(vi) We show that if X = L,*(X) then X = LU*(£) whence L,* = LU* since

we already have Lt* g LU* . Suppose now that G e LU(X) and S£ is the set of 3£-subgroups of G. Now since X = tt*(X), u & e 3C for any nonempty tower 3~ s 3C. Since G e LU(X) there exists an upper-directed local system 2. for G with J2 S 3C. Now by Zorn's Lemma G = u ief and so GeX. Hence LU(X) = X and so Lu*(£) = 3C, as required.

4. The class £s We begin with

PROOF OF THEOREM B. Suppose that a group G has a weakly upper-directed

local system 2. of £s-subgroups, Jf is a set of serial subgroups of G and J = <^f > = . We show first that J n Q ser Q for all g e J, i.e. Jn Q is a serial subgroup of Q.

Let ge2. and jceJnQ ^ J. Then there exist Hl,--,HneJ? and finitely generated subgroups Ht* of each Ht such that xe^Hl*,---,Hn*y. Now since J is weakly upper-directed there exists Re2 such that

g/i. Let

iC = (HlnR,-,HnnR}. Now HjCiRserR for 1 ^ i ^ n and Refi,, so Xseri? and so, since Q ^ R,

KnQ ser Q. But, since Ht* ^ H; n R ^ J, xeXnQ ^ J ng. Thus JOg is the join of all its subgroups which are serial in Q e 2S and so J n Q ser g. Hence

by Lemma 1 of [2], J ser G. Thus LW(£S) = £s and so LW*(£S) = fis.

Finally if X = sX s £s then L£ = LU3£ and SOLJC^. We will require one further lemma for the proof of Theorem C.

LEMMA 4.1. Let 3f be a set of finite ascendant subgroups of a group G. Then G PROOF. If J = <-?f > and K = then K is generated by finite ascendant subgroups of G and so K is locally finite. Thus by Theorem B or Theorem A of [2], Ke2s and so J ser K. But K is normal in G and so J ser G, as required. 1 PROOF OF THEOREM C. Suppose that GeSDt*, M is a set of serial subgroups of G and J = <^>. Then each H e Jf is necessarily an ascendant subgroup of G and there exists a normal subgroup H* of finite index in H and having no finite quotient groups. Then by Lemma 4.3 of [5] and its Corollary (cf. Lemma 4 of [6]), J* o -d G where J* = <#*|tfe JQ. Suppose now that L = and M = . Then M

References

[1] P. Hall, 'On non-strictly simple groups', Proc. Cambridge Philos. Soc. 59 (1963), 531-553. [2] B. Hartley, 'Serial subgroups of locally finite groups', Proc. Cambridge Philos. Soc. 71 (1972), 199-201. [3] K. K. Hickin, 'A class of groups whose local sequence is nonstationary', Proc. Amer. Math. Soc. 21 (1969), 402-408. [4] A. G. Kuros, The Theory of Groups, Vol. 2 (Chelsea, New York, 1956). [5] D. J. S. Robinson, 'On the theory of subnormal subgroups', Math. Zeit. 89 (1965), 30-51. [6] J. E. Roseblade, 'On certain subnormal coalition classes',/. Algebra 1 (1964), 132-138. [7] W. Sierpinski, Cardinal and Ordinal Numbers (Monografie Matematyczne, Tom 34, Polish Scientific Publishers, Warsaw, 2nd Ed. Revised 1965). [8] H. Zassenhaus, The Theory of Croups, (Chelsea, New York, 2nd Ed. 1958).

Mathematical Institute University of Edinburgh Scotland.